Gyrokinetic field theory

The Lagrangian formulation of the gyrokinetic theory is generalized in order to describe the particles\u27 dynamics, as well as the self-consistent behavior of the electromagnetic fields. The gyrokinetic equation for the particle distribution function and the gyrokinetic Maxwell\u27s equations, for the electromagnetic fields, are both derived from the variational principle for the Lagrangian consisting of the parts of particles, fields, and their interaction. In this generalized Lagrangian formulation, the energy conservation property for the total nonlinear gyrokinetic system of equations is directly shown from Noether\u27s theorem. This formulation can be utilized in order to derive the nonlinear gyrokinetic system of equations and the rigorously conserved total energy for fluctuations with arbitrary frequencies. Simplified gyrokinetic systems of equations with the conserved energy are obtained from the Lagrangian with the small electron gyroradii, quasineutrality, and linear polarization?magnetization approximations